Find the interval/s in which the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = xe^x$, is increasing.

$[-1, \infty)$

The correct answer is Option (2) → $[-1, \infty)$ ##

We have, $f(x) = xe^x$

$\Rightarrow f'(x) = e^x(x + 1)$

For $f(x)$ to be increasing, we have $f'(x) = e^x(x + 1) \geq 0$

$\Rightarrow x \geq -1$ as $e^x > 0, \forall x \in \mathbb{R}$

Hence, the required interval where $f(x)$ increases is $[-1, \infty)$.